Increments of the renewal function related to the distributions with infinite means and regularly varying tails of orders α ∈ (0,1] were described by Erickson in 1979 (Trans. Amer. Math. Soc. 151: 263–291, 1970). However, explicit asymptotics for the increments are known for α ∈ (½,1] only. For smaller α one can get, generally speaking, only the lower limit of the increments. There are many examples showing that this statement cannot be improved in general.
We refine Erikson's results by describing sufficient conditions for regularity of the renewal measure density of the distributions with regularly varying tails with α ∈ (0,½]. We also discuss the reasons of non-regular behavior of the renewal function increments in the general situation.